The absolute phase effect in energy model

Abstract

The energy model for motion and binocular vision has been considered to be invariant with respect to the absolute phases of the input stimuli¹, ². This has not been disputed since the introduction of the energy model (pointed out by an anonymous reviewer). However, Zhao and Farell ³ reported that the binocular energy model may depend on the absolute phases. Yet, many consider the result by Zhao and Farell ³ is due a bug in the simulation. Here we present an analytic proof showing that the energy model does depend on the absolute phases of the input stimuli.

For a sinusoidal stimulus oriented parallel to the major axis of the receptive field of complex cells, the model yields a response for these cells that is the sum of three parts: (1) cos² {[(p ¹_r−p ²_r)+( p ¹_s−p ²_s)]/2} and a constant. Here p ¹_r and p ²_r are the absolute phases of the receptive fields, p ¹_s and p ²_s are the absolute phases of the input stimuli. (2) cos² {[(p ¹_r−p ²_r)+( p ¹_s−p ²_s)]/2} and a constant; (3) cos( p ¹_s+p ²_s) [(cos(p ¹_r−p ²_r)+ cos( p ¹_s−p ²_s)] and a constant. Clearly, the first and the second terms do not depend on the absolute phases of the input stimuli since (p ¹_s−p ²_s) is the relative phase of the input stimuli. However, because of cos(p ¹_s+p ²_s), the third term depends on the absolute phases.

This theoretical result is analytical without any approximation—This is fundamentally different from former research on the analysis of the energy model. Therefore the analytical result should give every detail of the information of the model. Furthermore, the proof can be easily generalized to related stimuli, such as gabor patches.

The proof also holds for the motion energy model, since it is a special case of the binocular energy model.

References:

1. E.H Adelson, J. Bergen (1985) JOSA A, 2, 284.

2. I. Ohzawa, G.C. DeAngelis, R.D. Freeman. (1990) Science, 249, 1037.

3. L. Zhao, B. Farell. (2002) Journal of Vision. 2(7), 312.